A fundamental concept in quantum information theory is the understanding of
entanglement. Quantum entanglement can be viewed as a crucial resource in
quantum information. The key question is how to quantify and classify
entanglement of quantum states. Polynomial functions in the coefficients of
pure states which are invariant under stochastic local operations and
classical communication (SLOCC) transformations have been studied extensively
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}
and exploited to construct entanglement measures
[1]}, [2]}, [5]}, [12]}, [13]}, [14]}. The concurrence
[1]} and three-tangle [2]}, which measure entanglement
of two-qubit and three-qubit states, are polynomial invariants of degrees 2
and 4 respectively. It is known that the concurrence and three-tangle are
the absolute values of hyperdeterminants for two and three qubits
respectively [3]}.
An expression has recently been derived for four-tangle, which is
a polynomial invariant and a measure of genuine entanglement of four-qubit
states [4]}.
Polynomial invariants of degrees 2, 4 and 6 for four and
five qubits have been constructed from classical invariant theory [5]}, [6]}. The absolute values of the polynomial invariants obtained
in [5]} may be used to construct entanglement measures of
four-qubit states. Further, polynomial invariants of degrees 2, 4, 6, 8, 10
and 12 for four and five qubits have been obtained using local invariant
operators [7]}. Despite these efforts, few attempts have so far
been made towards the generalization to higher number of qubits.
Three-tangle has been generalized to \(n\) -tangle for even \(n\)  qubits [8]} and has been shown to be equal to the square of the polynomial
invariant of degree 2 [9]}. A generalization of three-tangle to odd \(n\)  qubits has been recently proposed in [10]}. In [11]},
polynomial invariants of degree 2 for even \(n\)  qubits and degree 4 for odd \(n\)  qubits have been derived by induction based on the definition of SLOCC.
